In x-ray computed tomography, the object from which projection data is to be acquired is generally exposed to x-ray radiation from a number of projection directions. An image dataset is then reconstructed from the projection data. This is generally done using a back-projection method, wherein the projection data is often pre-processed. For example, a so-called rebinning step is often performed, in which the projection data generated from a fan-shaped beam of x-rays is rearranged, so that it is in such a form as if the x-ray beams had been parallel. The data that has been thus rearranged and filtered is then used for a back-projection onto individual image points or voxels within the volume of interest.
One standard method generally used for the reconstruction is a so-called filtered back-projection method (FBP). With this method, the rebinned data is generally first transformed into the frequency range, where filtering takes place by multiplication using a convolution kernel. The filtered data is then back-transformed and the back-projection is done on the filtered data. The selection of the convolution kernel allows the desired image characteristics, in particular image sharpeners and noise, to be influenced. One algorithm often used is the so-called Feldkamp algorithm.
However, simple back-projection methods have the disadvantage that they do not produce satisfactory results when the projection data is incomplete, for example when the projection directions do not cover an angle of 180° plus the fan beam angle around the object, or the object is not fully within the field-of-view of all projections. They also are not able to fully account for statistical properties of the data.
Other reconstructions methods which have more recently been developed are iterative image reconstruction methods. In such an iterative reconstruction method, a reconstruction of initial image data from the measured projection data takes place first. From this initial image data, a “projector” (projection operator), which should map the projection geometry of the CT-system and the data acquisition process mathematically as closely as possible, is then used to generate synthetic projection data. The difference in respect of the measured projection data is then back-projected, thereby reconstructing a residue image, which can be used to update the initial image. The updated image data can in turn be used to generate new synthetic projection data in a next iteration step with the aid of a projection operator, to form the difference to the measurement signals again and to calculate a new residue image, which can in turn be used to improve the image data of the current iteration stage.
However, these iterative image reconstruction algorithms often do not converge fast, and are therefore very computation-intensive. Several reconstruction algorithms based on the penalized weighted least-square model have been suggested. However, no iterative reconstruction method is currently considered as satisfactory for routine clinical usage.
The so-called FISTA (fast iterative shrinkage-thresholding algorithm) is one algorithm for statistical iterative reconstruction methods for x-ray CT. It updates all voxels simultaneously with a fast outer converging loop. However, the method is hampered by the need to solve a non-linear image denoising problem with correlated voxels at each iteration. FISTA is described in the paper by Amir Beck and Marc Teboulle “A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems” SIAM, J. Imaging Sciences, Vol. 2, No. 1, pp. 183-202. The content of this paper is hereby incorporated herein by reference in its entirety.